Eamcet - Maths - Theory Of Equations

If the roots of the equation 4x3-12x2+11 x + k = 0 are in arithmetic progression, then k is equal to :

A.  -3

B.  1

C.  2

D.  3

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If f(x) is a polynomial of degree n with rational coefficients and 1+2i, 2-√3 and 5 are three roots of       f(x) = 0, then the least value of n is :

A.  5

B.  4

C.  3

D.  6

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If cos 2x = (√2 + 1)(cos x - 1/√2) , cos x ≠ 1/2 then x belongs to

A.  {2nπ ± π/6 : n Є Z }

B.  {2nπ ±π/3 : n Є Z }

C.  {2nπ ±π/2 : n Є Z }

D.  {2nπ ± π/4 : n Є Z }

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The sum of the fourth powers of the roots of the equation x5+px3+qx2+s=0 is

A.  2p2

B.  3p2

C.  2p

D.  3p

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If the roots of x4+5x3-30x2-40x+64=0 are in G.P then the roots are

A.   1,-2,4,-8

B.  ±1, 2,3

C.  ±2i, 2 3

D.  -3/2,-1/3,2±√3

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If A,B,C are the remainders of x3-3x2-x+5,3x4-x3+2x2-2x-4,2x5-3x4+5x3-7x2+3x-4 when divided by x+1,x+2,x-2 respectively then the ascending order of A,B,C is

A.  A,B,C

B.  B,C,A

C.  A,C,B

D.  B,A,C

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If α and β are different values of x satisfying a cos x+b sin x=c then tan (α+β/2) =

A.  a+b

B.  a-b

C.   a/b

D.  b/a

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If x2+p1x+q1=0,x2+p2x+q2=0,x2+p3x+q3=0 has a common root,then p12+p22+p32+4(q1+q2+q3)=

A.  2(p1p2+p2p3+p3p1)

B.  (p2p1+q2p3+q3p1)

C.  2(q1p2+q2p3+p3q1)

D.  none

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The equation whose roots are multiplied by 3 of those of 2x2+3x-1=0 is

A.  2x2+9x-9=0

B.  2x2-7x+6=0

C.  2x2+5x+7=0

D.  3x2+5x-7=0

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The roots of x3-3x2+4=0,when there is a multiple root,are

A.  1/2,1,-3

B.  1/3,1/3,1

C.  6,4,-1

D.  2,2,-1

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The equations whose roots are exceed by 2 than those of x4+x3-10x2+4x+24=0 is

A.  x4-7x3+8x2+24x-16=0

B.  x5-7x3+12x2-7x=0

C.  x4+7x3-11x2-144x-208=0

D.  x5+11x4+45x3+81x2+50x-6=0

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If 2,-2,4 are the roots of ax3+bx2+cx+d=0 then the roots of 8ax3+4bx2+2cx+d=0 are

A.  2,-2,4

B.  1/2,-1/2,1/4

C.  1,-1,2

D.  4,-4,8

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If α+β+γ=1,α2+β2+γ2=2 and α3+β3+γ3=3,then α5+β5+γ5=

A.  6

B.  18

C.  36

D.  45

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If the equations x2-x-p=0 and x2+2px-12=0 have a common root,then that root is

A.  i

B.  p+2

C.  2

D.  cannot be determined

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If a=cos 2π/7+i sin 2π/7,α=a+a2+a4 and β=a3+a5+a6 then α,β are the roots of the equation

A.  x2+x+1=0

B.  x2+x+2=0

C.  x2+2x+2=0

D.  x2+2x+3=0

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The values of the parameters a for which the quadratic equations (1-2a)x2-6ax-1=0 and ax2-x+1=0 have at least one root in common are

A.  0,1/2

B.  1/2,2/9

C.  2/9

D.  0,1/2,2/9

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The roots of x4-5x3+3x2+19x-30=0 are

A.  -2,3,3±i

B.  -2,3,2±3i

C.  -2,3,2±i

D.  -2,3,3±2i

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If α,β are the roots of 8x2-3x+27=0 then the value of (α2/β)1/3+(β2/α)1/3 is

A.  1/3

B.  1/4

C.  7/2

D.  4

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The roots of x3+x2-2x-2=0 are

A.  -1,±√2

B.  0,1,2

C.  -2,-1,1

D.  1,±i

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The transformed equation of x3-4x2+1/4x-1/9=0,by eliminating fractional coefficients is

A.  y3+15y2+52y-36=0

B.  y4-24y2+65y-55=0

C.  4x4-2x3+6x2-3x-1=0

D.  y3-24y2+9y-24=0

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The remainder when 2x5-3x4+5x3-3 x2+7x-9 with x2-x-3 is

A.  33x+4

B.  41x+3

C.  41x+44

D.  33x-4

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The equations whose roots are diminish by 3 than those of x4-5x3-20x2+3x+17=0 is

A.  x4-9x3+20x2=0

B.  x5-7x3+12x2-7x=0

C.  x4+7x3-11x2-144x-208=0

D.  x5+11x4+45x3+81x2+50x-6=0

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If α,β,γ are the roots of x3+x2+x+1=0 then α3+β3+γ3=

A.  1

B.  -1

C.  2

D.  -2

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If one root of the equation ix2-2(1+i)x+(2-i)=0 is 2-I,then the other root is

A.  -i

B.  2+i

C.  i

D.  2-i

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Match the following. Equation Roots I.x3-3x2-16x+48 =0 a)6,4,-1 II.x3-7x2+14x-8=0 b)1,1/3,1/5 III.15x3-23x2-9x-1=0 c)1,2,4 IV. x3-9x2+14x+24=0 d)4,-4,3

A.  c, d, a, b

B.  d ,c, b, a

C.  c, a, b, d

D.  c, b, a, d

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If α,β,γ are the roots of x3+3px+q=0 then the equation whose roots are α+1/β+γ–α,β+1/γ+α–β and γ+1/α+β–γ is

A.  8y3+12y3+(6+6p) y + 1 +3p –q=0

B.  8qy3-12(q+ p)y2 + 6(q-2p) y + (q+3p-1)=0

C.  8qy3+12(q+ p)y2 + 6(q-2p) y + (q+3p-1)=0

D.  8qy3-12(q+ p)y2 -6(q-2p) y + (q+3p-1)=0

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If the quadratic equations ax2+2cx+b=0 and ax2+2bx+c=0 (b≠c) have a common root,then a+4b+4c=

A.  -2

B.  -1

C.  0

D.  1

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If α,β,γ are the roots of x3+2x2-3x-1=0,then the value of α-4+β-4+γ-4 is

A.  123

B.  36

C.  149

D.  795

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If A12/x-a1+A22/x-a2+.... Ak2/x-a k=m and ai,AI,m are rational then the equation has

A.  no imaginary roots

B.  no positive roots

C.  no negative roots

D.  no real roots

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If the sum of two of the roots of 4x3+16x2-9x-36=0 is zero then the roots are

A.  ±√5, 1+i

B.  3/2,-3/2,-4

C.  -1/2, 1/2,-1/5

D.  ±√2, √5

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